Patchworking real algebraic varieties

TL;DR

Patchworking is a method for constructing real algebraic hypersurfaces by gluing simpler pieces, enabling the creation of complex topologies and connecting to tropical geometry, with broad applications in algebraic geometry.

Contribution

This paper presents the original, full generality of the patchworking technique, expanding its theoretical foundation and applications in real algebraic geometry.

Findings

Constructed real algebraic hypersurfaces with prescribed topology.

Connected patchworking to tropical geometry via combinatorial methods.

Provided a comprehensive presentation of the patchworking technique.

Abstract

Patchworking is a construction of a one-parameter family of real algebraic hypersurfaces. For sufficiently small positive values of the parameter, the hypersurfaces can be obtained by gluing of given hypersurfaces topologically. The author invented patchworking in 1979-81 and used it for constructing of real plane algebraic curves with complicated prescribed topology. In particular, it helped to complete isotopy classification of nonsingular plane projective real algebraic curves of degree 7. A special case of the patchworking, combinatorial patchworking, can be considered as Litvinov-Maslov quantization of a tropical variety. Due to its simplicity, combinatorial patchworking is better known than the general one. This paper is the original presentation of the patchworking, in its full generality.